A381380 Decimal expansion of the area of a ruled surface formed by moving a segment of length sqrt(6), the ends of which lie on the diagonals of opposite faces of a unit cube oriented at right angles to each other.

2, 7, 2, 7, 0, 5, 4, 7, 7, 3, 8, 1, 2, 0, 4, 8, 9, 8, 8, 4, 3, 5, 1, 5, 5, 6, 7, 9, 0, 2, 0, 2, 5, 9, 8, 4, 2, 8, 3, 4, 6, 4, 7, 7, 1, 9, 9, 0, 3, 1, 3, 8, 7, 4, 0, 0, 3, 1, 0, 7, 1, 1, 8, 9, 3, 9, 5, 3, 9, 5, 1, 4, 0, 1, 3, 6, 7, 1, 4, 8, 4, 8, 4, 4, 9, 4, 0, 4, 0, 1, 1

Offset: 1

Author

Nicolay Avilov, Feb 22 2025

Keywords

• nonn
• cons

Comments

A segment of constant length continuously sliding its endpoints along two intersecting straight lines defines a ruled surface -- a linoid. Here we consider a linoid defined by a segment of length sqrt(6)/2 sliding along two intersecting diagonals of opposite faces of a cube with edge 1. The surface of a linoid is given by the equation 2*x^2/(z - 1/2)^2 + 2*y^2/(z + 1/2)^2 = 1.

The surface of a linoid consists of four congruent surfaces. The area of one of them is calculated using integrals and multiplied by 4.

The name of the figure "linoid" was introduced by the author in the related article, see link.

Examples

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Links

• Nicolay Avilov, Construction of a linoid
• Nicolay Avilov, Volume of a "linoid" (in Russian).

Formula

Equals sqrt(2)*Integral_{t=0..Pi/2} Integral_{z=0..1/2} sqrt(5 + 24*z^2 + 24*z*cos(2*t) + cos(4*t)) dz dt.

Extensions

Terms corrected by Jinyuan Wang, Feb 23 2025